2.30.24 Problem 133

Internal problem ID [13794]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-4
Problem number : 133
Date solved : Friday, December 19, 2025 at 12:38:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+a y^{\prime } x +x^{n} \left (b \,x^{n}+c \right ) y&=0 \\ \end{align*}

2.30.24.1 ✓ Maple. Time used: 0.020 (sec). Leaf size: 82

ode:=x^2*diff(diff(y(x),x),x)+a*x*diff(y(x),x)+x^n*(b*x^n+c)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (\operatorname {WhittakerM}\left (-\frac {i c}{2 n \sqrt {b}}, \frac {a -1}{2 n}, \frac {2 i \sqrt {b}\, x^{n}}{n}\right ) c_1 +\operatorname {WhittakerW}\left (-\frac {i c}{2 n \sqrt {b}}, \frac {a -1}{2 n}, \frac {2 i \sqrt {b}\, x^{n}}{n}\right ) c_2 \right ) x^{-\frac {a}{2}+\frac {1}{2}-\frac {n}{2}} \]

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying a symmetry of the form [xi=0, eta=F(x)] 
checking if the LODE is missing y 
-> Trying an equivalence, under non-integer power transformations, 
   to LODEs admitting Liouvillian solutions. 
   -> Trying a Liouvillian solution using Kovacics algorithm 
   <- No Liouvillian solutions exists 
-> Trying a solution in terms of special functions: 
   -> Bessel 
   -> elliptic 
   -> Legendre 
   -> Whittaker 
      -> hyper3: Equivalence to 1F1 under a power @ Moebius 
      <- hyper3 successful: received ODE is equivalent to the 1F1 ODE 
   <- Whittaker successful 
<- special function solution successful
 

2.30.24.2 ✓ Mathematica. Time used: 0.089 (sec). Leaf size: 165

ode=x^2*D[y[x],{x,2}]+a*x*D[y[x],x]+x^n*(b*x^n+c)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to 2^{\frac {a+n-1}{2 n}} x^{\frac {1}{2} (-a-n+1)} \left (x^n\right )^{\frac {a+n-1}{2 n}} e^{\frac {i \sqrt {b} x^n}{n}} \left (c_1 \operatorname {HypergeometricU}\left (-\frac {-a+\frac {i c}{\sqrt {b}}-n+1}{2 n},\frac {a+n-1}{n},-\frac {2 i \sqrt {b} x^n}{n}\right )+c_2 L_{-\frac {a-\frac {i c}{\sqrt {b}}+n-1}{2 n}}^{\frac {a-1}{n}}\left (-\frac {2 i \sqrt {b} x^n}{n}\right )\right ) \end{align*}

2.30.24.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
n = symbols("n") 
y = Function("y") 
ode = Eq(a*x*Derivative(y(x), x) + x**2*Derivative(y(x), (x, 2)) + x**n*(b*x**n + c)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None
 

Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0] 
Sympy version 1.14.0
 

classify_ode(ode,func=y(x)) 
 
('factorable', '2nd_power_series_regular')